Influence of ideals in compactifications

TL;DR

This paper explores the properties of one point compactifications influenced by ideals of subsets of natural numbers, introducing the concept of -proper maps and conditions for their extension.

Contribution

It introduces -proper maps, the shrinking condition (C), and characterizes when -compactifications coincide with classical compactifications under certain ideal classes.

Findings

-proper maps can be extended to -compactifications

The shrinking condition (C) is crucial for studying -proper maps

Certain ideal classes ensure -compactification matches classical compactification in metrizable spaces

Abstract

One point compactification is studied in the light of ideal of subsets of $\mathbb{N}$. $\mathcal{I}$-proper map is introduced and showed that a continuous map can be extended continuously to the one point $\mathcal{I}$-compactification if and only if the map is $\mathcal{I}$-proper. Shrinking condition(C) introduced in this article plays an important role to study various properties of $\mathcal{I}$-proper maps. It is seen that one point $\mathcal{I}$-compactification of a topological space may fail to be Hausdorff but a class $\{\mathcal{I}\}$ of ideals has been identified for which one point $\mathcal{I}$-compactification coincides with the one point compactification if it is metrizable.